(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo1.
Require Import Ranalysis1.
Require Import PSeries_reg.
Local Open Scope nat_scope.
Local Open Scope R_scope.


(**********)
Lemma continuity_sin : continuity sin.
continuity sin
Proof.
continuity sin
  unfold continuity; intro.
x:R
continuity_pt sin x
  assert (H0 := continuity_cos (PI / 2 - x)).
x:R
H0:continuity_pt cos (PI / 2 - x)
continuity_pt sin x
  unfold continuity_pt in H0; unfold continue_in in H0; unfold limit1_in in H0;
    unfold limit_in in H0; simpl in H0; unfold R_dist in H0;
      unfold continuity_pt; unfold continue_in;
        unfold limit1_in; unfold limit_in;
          simpl; unfold R_dist; intros.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond (PI / 2 - x) x0 /\
   Rabs (x0 - (PI / 2 - x)) < alp ->
   Rabs (cos x0 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (sin x0 - sin x) < eps)
  elim (H0 _ H); intros.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond (PI / 2 - x) x1 /\
   Rabs (x1 - (PI / 2 - x)) < alp ->
   Rabs (cos x1 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x1 : R,
 D_x no_cond (PI / 2 - x) x1 /\
 Rabs (x1 - (PI / 2 - x)) < x0 ->
 Rabs (cos x1 - cos (PI / 2 - x)) < eps)
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond x x1 /\ Rabs (x1 - x) < alp ->
   Rabs (sin x1 - sin x) < eps)
  exists x0; intros.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond (PI / 2 - x) x1 /\
   Rabs (x1 - (PI / 2 - x)) < alp ->
   Rabs (cos x1 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x1 : R,
 D_x no_cond (PI / 2 - x) x1 /\
 Rabs (x1 - (PI / 2 - x)) < x0 ->
 Rabs (cos x1 - cos (PI / 2 - x)) < eps)
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (sin x1 - sin x) < eps)
  elim H1; intros.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond (PI / 2 - x) x1 /\
   Rabs (x1 - (PI / 2 - x)) < alp ->
   Rabs (cos x1 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x1 : R,
 D_x no_cond (PI / 2 - x) x1 /\
 Rabs (x1 - (PI / 2 - x)) < x0 ->
 Rabs (cos x1 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x1 : R,
D_x no_cond (PI / 2 - x) x1 /\
Rabs (x1 - (PI / 2 - x)) < x0 ->
Rabs (cos x1 - cos (PI / 2 - x)) < eps
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (sin x1 - sin x) < eps)
  split.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond (PI / 2 - x) x1 /\
   Rabs (x1 - (PI / 2 - x)) < alp ->
   Rabs (cos x1 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x1 : R,
 D_x no_cond (PI / 2 - x) x1 /\
 Rabs (x1 - (PI / 2 - x)) < x0 ->
 Rabs (cos x1 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x1 : R,
D_x no_cond (PI / 2 - x) x1 /\
Rabs (x1 - (PI / 2 - x)) < x0 ->
Rabs (cos x1 - cos (PI / 2 - x)) < eps
x0 > 0
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond (PI / 2 - x) x1 /\
   Rabs (x1 - (PI / 2 - x)) < alp ->
   Rabs (cos x1 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x1 : R,
 D_x no_cond (PI / 2 - x) x1 /\
 Rabs (x1 - (PI / 2 - x)) < x0 ->
 Rabs (cos x1 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x1 : R,
D_x no_cond (PI / 2 - x) x1 /\
Rabs (x1 - (PI / 2 - x)) < x0 ->
Rabs (cos x1 - cos (PI / 2 - x)) < eps
forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
Rabs (sin x1 - sin x) < eps
  assumption.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond (PI / 2 - x) x1 /\
   Rabs (x1 - (PI / 2 - x)) < alp ->
   Rabs (cos x1 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x1 : R,
 D_x no_cond (PI / 2 - x) x1 /\
 Rabs (x1 - (PI / 2 - x)) < x0 ->
 Rabs (cos x1 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x1 : R,
D_x no_cond (PI / 2 - x) x1 /\
Rabs (x1 - (PI / 2 - x)) < x0 ->
Rabs (cos x1 - cos (PI / 2 - x)) < eps
forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
Rabs (sin x1 - sin x) < eps
  intros; rewrite <- (cos_shift x); rewrite <- (cos_shift x1); apply H3.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond (PI / 2 - x) x2 /\
   Rabs (x2 - (PI / 2 - x)) < alp ->
   Rabs (cos x2 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x2 : R,
 D_x no_cond (PI / 2 - x) x2 /\
 Rabs (x2 - (PI / 2 - x)) < x0 ->
 Rabs (cos x2 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x2 : R,
D_x no_cond (PI / 2 - x) x2 /\
Rabs (x2 - (PI / 2 - x)) < x0 ->
Rabs (cos x2 - cos (PI / 2 - x)) < eps
x1:R
H4:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
D_x no_cond (PI / 2 - x) (PI / 2 - x1) /\
Rabs (PI / 2 - x1 - (PI / 2 - x)) < x0
  elim H4; intros.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond (PI / 2 - x) x2 /\
   Rabs (x2 - (PI / 2 - x)) < alp ->
   Rabs (cos x2 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x2 : R,
 D_x no_cond (PI / 2 - x) x2 /\
 Rabs (x2 - (PI / 2 - x)) < x0 ->
 Rabs (cos x2 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x2 : R,
D_x no_cond (PI / 2 - x) x2 /\
Rabs (x2 - (PI / 2 - x)) < x0 ->
Rabs (cos x2 - cos (PI / 2 - x)) < eps
x1:R
H4:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H5:D_x no_cond x x1
H6:Rabs (x1 - x) < x0
D_x no_cond (PI / 2 - x) (PI / 2 - x1) /\
Rabs (PI / 2 - x1 - (PI / 2 - x)) < x0
  split.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond (PI / 2 - x) x2 /\
   Rabs (x2 - (PI / 2 - x)) < alp ->
   Rabs (cos x2 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x2 : R,
 D_x no_cond (PI / 2 - x) x2 /\
 Rabs (x2 - (PI / 2 - x)) < x0 ->
 Rabs (cos x2 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x2 : R,
D_x no_cond (PI / 2 - x) x2 /\
Rabs (x2 - (PI / 2 - x)) < x0 ->
Rabs (cos x2 - cos (PI / 2 - x)) < eps
x1:R
H4:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H5:D_x no_cond x x1
H6:Rabs (x1 - x) < x0
D_x no_cond (PI / 2 - x) (PI / 2 - x1)
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond (PI / 2 - x) x2 /\
   Rabs (x2 - (PI / 2 - x)) < alp ->
   Rabs (cos x2 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x2 : R,
 D_x no_cond (PI / 2 - x) x2 /\
 Rabs (x2 - (PI / 2 - x)) < x0 ->
 Rabs (cos x2 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x2 : R,
D_x no_cond (PI / 2 - x) x2 /\
Rabs (x2 - (PI / 2 - x)) < x0 ->
Rabs (cos x2 - cos (PI / 2 - x)) < eps
x1:R
H4:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H5:D_x no_cond x x1
H6:Rabs (x1 - x) < x0
Rabs (PI / 2 - x1 - (PI / 2 - x)) < x0
  unfold D_x, no_cond; split.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond (PI / 2 - x) x2 /\
   Rabs (x2 - (PI / 2 - x)) < alp ->
   Rabs (cos x2 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x2 : R,
 D_x no_cond (PI / 2 - x) x2 /\
 Rabs (x2 - (PI / 2 - x)) < x0 ->
 Rabs (cos x2 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x2 : R,
D_x no_cond (PI / 2 - x) x2 /\
Rabs (x2 - (PI / 2 - x)) < x0 ->
Rabs (cos x2 - cos (PI / 2 - x)) < eps
x1:R
H4:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H5:D_x no_cond x x1
H6:Rabs (x1 - x) < x0
True
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond (PI / 2 - x) x2 /\
   Rabs (x2 - (PI / 2 - x)) < alp ->
   Rabs (cos x2 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x2 : R,
 D_x no_cond (PI / 2 - x) x2 /\
 Rabs (x2 - (PI / 2 - x)) < x0 ->
 Rabs (cos x2 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x2 : R,
D_x no_cond (PI / 2 - x) x2 /\
Rabs (x2 - (PI / 2 - x)) < x0 ->
Rabs (cos x2 - cos (PI / 2 - x)) < eps
x1:R
H4:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H5:D_x no_cond x x1
H6:Rabs (x1 - x) < x0
PI / 2 - x <> PI / 2 - x1
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond (PI / 2 - x) x2 /\
   Rabs (x2 - (PI / 2 - x)) < alp ->
   Rabs (cos x2 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x2 : R,
 D_x no_cond (PI / 2 - x) x2 /\
 Rabs (x2 - (PI / 2 - x)) < x0 ->
 Rabs (cos x2 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x2 : R,
D_x no_cond (PI / 2 - x) x2 /\
Rabs (x2 - (PI / 2 - x)) < x0 ->
Rabs (cos x2 - cos (PI / 2 - x)) < eps
x1:R
H4:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H5:D_x no_cond x x1
H6:Rabs (x1 - x) < x0
Rabs (PI / 2 - x1 - (PI / 2 - x)) < x0
  trivial.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond (PI / 2 - x) x2 /\
   Rabs (x2 - (PI / 2 - x)) < alp ->
   Rabs (cos x2 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x2 : R,
 D_x no_cond (PI / 2 - x) x2 /\
 Rabs (x2 - (PI / 2 - x)) < x0 ->
 Rabs (cos x2 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x2 : R,
D_x no_cond (PI / 2 - x) x2 /\
Rabs (x2 - (PI / 2 - x)) < x0 ->
Rabs (cos x2 - cos (PI / 2 - x)) < eps
x1:R
H4:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H5:D_x no_cond x x1
H6:Rabs (x1 - x) < x0
PI / 2 - x <> PI / 2 - x1
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond (PI / 2 - x) x2 /\
   Rabs (x2 - (PI / 2 - x)) < alp ->
   Rabs (cos x2 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x2 : R,
 D_x no_cond (PI / 2 - x) x2 /\
 Rabs (x2 - (PI / 2 - x)) < x0 ->
 Rabs (cos x2 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x2 : R,
D_x no_cond (PI / 2 - x) x2 /\
Rabs (x2 - (PI / 2 - x)) < x0 ->
Rabs (cos x2 - cos (PI / 2 - x)) < eps
x1:R
H4:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H5:D_x no_cond x x1
H6:Rabs (x1 - x) < x0
Rabs (PI / 2 - x1 - (PI / 2 - x)) < x0
  red; intro; unfold D_x, no_cond in H5; elim H5; intros _ H8; elim H8;
    rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive x1);
      apply Ropp_eq_compat; apply Rplus_eq_reg_l with (PI / 2);
        apply H7.
x:R
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond (PI / 2 - x) x2 /\
   Rabs (x2 - (PI / 2 - x)) < alp ->
   Rabs (cos x2 - cos (PI / 2 - x)) < eps0)
eps:R
H:eps > 0
x0:R
H1:x0 > 0 /\
(forall x2 : R,
 D_x no_cond (PI / 2 - x) x2 /\
 Rabs (x2 - (PI / 2 - x)) < x0 ->
 Rabs (cos x2 - cos (PI / 2 - x)) < eps)
H2:x0 > 0
H3:forall x2 : R,
D_x no_cond (PI / 2 - x) x2 /\
Rabs (x2 - (PI / 2 - x)) < x0 ->
Rabs (cos x2 - cos (PI / 2 - x)) < eps
x1:R
H4:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H5:D_x no_cond x x1
H6:Rabs (x1 - x) < x0
Rabs (PI / 2 - x1 - (PI / 2 - x)) < x0
  replace (PI / 2 - x1 - (PI / 2 - x)) with (x - x1); [ idtac | ring ];
  rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply H6.
Qed.

Lemma CVN_R_sin :
  forall fn:nat -> R -> R,
    fn =
    (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)) ->
    CVN_R fn.
forall fn : nat -> R -> R,
fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)) ->
CVN_R fn
Proof.
forall fn : nat -> R -> R,
fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)) ->
CVN_R fn
  unfold CVN_R; unfold CVN_r; intros fn H r.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{An : nat -> R &
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun k : nat => Rabs (An k)) n) l /\
(forall (n : nat) (y : R),
 Boule 0 r y -> Rabs (fn n y) <= An n)}}
  exists (fun n:nat => / INR (fact (2 * n + 1)) * r ^ (2 * n)).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <=
 / INR (fact (2 * n + 1)) * r ^ (2 * n))}
  cut
    { l:R |
        Un_cv
        (fun n:nat =>
          sum_f_R0
          (fun k:nat => Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
        l }.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l} ->
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <=
 / INR (fact (2 * n + 1)) * r ^ (2 * n))}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  intros (x,p).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n) x
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <=
 / INR (fact (2 * n + 1)) * r ^ (2 * n))}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  exists x.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n) x
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  x /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <=
 / INR (fact (2 * n + 1)) * r ^ (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  split.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n) x
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  x
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n) x
forall (n : nat) (y : R),
Boule 0 r y ->
Rabs (fn n y) <=
/ INR (fact (2 * n + 1)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  apply p.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n) x
forall (n : nat) (y : R),
Boule 0 r y ->
Rabs (fn n y) <=
/ INR (fact (2 * n + 1)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  intros; rewrite H; unfold Rdiv; do 2 rewrite Rabs_mult;
    rewrite pow_1_abs; rewrite Rmult_1_l.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
Rabs (/ INR (fact (2 * n + 1))) * Rabs (y ^ (2 * n)) <=
/ INR (fact (2 * n + 1)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  cut (0 < / INR (fact (2 * n + 1))).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n + 1)) ->
Rabs (/ INR (fact (2 * n + 1))) * Rabs (y ^ (2 * n)) <=
/ INR (fact (2 * n + 1)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n + 1))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  intro; rewrite (Rabs_right _ (Rle_ge _ _ (Rlt_le _ _ H1))).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n + 1))
/ INR (fact (2 * n + 1)) * Rabs (y ^ (2 * n)) <=
/ INR (fact (2 * n + 1)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n + 1))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  apply Rmult_le_compat_l.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n + 1))
0 <= / INR (fact (2 * n + 1))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n + 1))
Rabs (y ^ (2 * n)) <= r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n + 1))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  left; apply H1.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n + 1))
Rabs (y ^ (2 * n)) <= r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n + 1))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  rewrite <- RPow_abs; apply pow_maj_Rabs.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n + 1))
Rabs (Rabs y) <= r
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n + 1))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  rewrite Rabs_Rabsolu; unfold Boule in H0; rewrite Rminus_0_r in H0; left;
    apply H0.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x0 ^ (2 * N))
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs
        (/ INR (fact (2 * k + 1)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n + 1))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  apply Rinv_0_lt_compat; apply INR_fact_lt_0.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
  cut ((r:R) <> 0).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0 ->
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
  l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  intro; apply Alembert_C2.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
forall n : nat,
Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs
        (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))))
  0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  intro; apply Rabs_no_R0.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
n:nat
/ INR (fact (2 * n + 1)) * r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs
        (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))))
  0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply prod_neq_R0.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
n:nat
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
n:nat
r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs
        (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))))
  0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rinv_neq_0_compat; apply INR_fact_neq_0.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
n:nat
r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs
        (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))))
  0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply pow_nonzero; assumption.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs
        (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))))
  0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  assert (H1 := Alembert_sin).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:Un_cv
  (fun n : nat => Rabs (sin_n (S n) / sin_n n)) 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs
        (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))))
  0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold sin_n in H1; unfold Un_cv in H1; unfold Un_cv; intros.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       (Rabs
          (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
        Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))))
    0 < eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  cut (0 < eps / Rsqr r).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r² ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       (Rabs
          (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
        Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))))
    0 < eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  intro; elim (H1 _ H3); intros N0 H4.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n : nat,
(n >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       (Rabs
          (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
        Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))))
    0 < eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  exists N0; intros.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
R_dist
  (Rabs
     (Rabs
        (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))))
  0 < eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold R_dist; assert (H6 := H4 _ H5).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs
  (Rabs
     (Rabs
        (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n))) -
   0) < eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold R_dist in H5;
    replace
    (Rabs
      (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
        Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))) with
    (Rsqr r *
      Rabs
      ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs
  (r² *
   Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1)))) - 0) < eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rmult_lt_reg_l with (/ Rsqr r).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
0 < / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ r² *
Rabs
  (r² *
   Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1)))) - 0) <
/ r² * eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ r² *
Rabs
  (r² *
   Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1)))) - 0) <
/ r² * eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  pattern (/ Rsqr r) at 1; rewrite <- (Rabs_right (/ Rsqr r)).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ r²) *
Rabs
  (r² *
   Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1)))) - 0) <
/ r² * eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ r² >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite <- Rabs_mult.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs
  (/ r² *
   (r² *
    Rabs
      ((-1) ^ S n / INR (fact (2 * S n + 1)) /
       ((-1) ^ n / INR (fact (2 * n + 1)))) - 0)) <
/ r² * eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ r² >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rmult_minus_distr_l.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs
  (/ r² *
   (r² *
    Rabs
      ((-1) ^ S n / INR (fact (2 * S n + 1)) /
       ((-1) ^ n / INR (fact (2 * n + 1))))) -
   / r² * 0) < / r² * eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ r² >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rmult_0_r; rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs
  (1 *
   Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1)))) - 0) <
/ r² * eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ r² >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rmult_1_l; rewrite <- (Rmult_comm eps).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1)))) - 0) <
eps * / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ r² >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply H6.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ r² >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold Rsqr; apply prod_neq_R0; assumption.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ r² >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rle_ge; left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n + 1)) /
   ((-1) ^ n / INR (fact (2 * n + 1)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold Rdiv; rewrite (Rmult_comm (Rsqr r)); repeat rewrite Rabs_mult;
    rewrite Rabs_Rabsolu; rewrite pow_1_abs.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
1 * Rabs (/ INR (fact (2 * S n + 1))) *
Rabs (/ ((-1) ^ n * / INR (fact (2 * n + 1)))) * r² =
Rabs (/ INR (fact (2 * S n + 1))) *
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n + 1))) *
    Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rmult_1_l.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * S n + 1))) *
Rabs (/ ((-1) ^ n * / INR (fact (2 * n + 1)))) * r² =
Rabs (/ INR (fact (2 * S n + 1))) *
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n + 1))) *
    Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  repeat rewrite Rmult_assoc; apply Rmult_eq_compat_l.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ ((-1) ^ n * / INR (fact (2 * n + 1)))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n + 1))) *
    Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rinv_mult_distr.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ (-1) ^ n * / / INR (fact (2 * n + 1))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n + 1))) *
    Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rinv_involutive.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ (-1) ^ n * INR (fact (2 * n + 1))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n + 1))) *
    Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_mult.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ (-1) ^ n) * Rabs (INR (fact (2 * n + 1))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n + 1))) *
    Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_Rinv.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ Rabs ((-1) ^ n) * Rabs (INR (fact (2 * n + 1))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n + 1))) *
    Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite pow_1_abs; rewrite Rinv_1; rewrite Rmult_1_l.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (INR (fact (2 * n + 1))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n + 1))) *
    Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rinv_mult_distr.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (INR (fact (2 * n + 1))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/ Rabs (/ INR (fact (2 * n + 1))) *
   / Rabs (r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite <- Rabs_Rinv.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (INR (fact (2 * n + 1))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (Rabs (/ / INR (fact (2 * n + 1))) *
   / Rabs (r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rinv_involutive.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (INR (fact (2 * n + 1))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (Rabs (INR (fact (2 * n + 1))) *
   / Rabs (r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_mult.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (INR (fact (2 * n + 1))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
(Rabs (Rabs (INR (fact (2 * n + 1)))) *
 Rabs (/ Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  do 2 rewrite Rabs_Rabsolu.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (INR (fact (2 * n + 1))) * r² =
Rabs (r ^ (2 * S n)) *
(Rabs (INR (fact (2 * n + 1))) *
 Rabs (/ Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite (Rmult_comm (Rabs (r ^ (2 * S n)))).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (INR (fact (2 * n + 1))) * r² =
Rabs (INR (fact (2 * n + 1))) *
Rabs (/ Rabs (r ^ (2 * n))) * Rabs (r ^ (2 * S n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rmult_assoc; apply Rmult_eq_compat_l.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² =
Rabs (/ Rabs (r ^ (2 * n))) * Rabs (r ^ (2 * S n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_Rinv.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² =
/ Rabs (Rabs (r ^ (2 * n))) * Rabs (r ^ (2 * S n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_Rabsolu.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² = / Rabs (r ^ (2 * n)) * Rabs (r ^ (2 * S n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  repeat rewrite Rabs_right.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² = / r ^ (2 * n) * r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  replace (r ^ (2 * S n)) with (r ^ (2 * n) * r * r).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² = / r ^ (2 * n) * (r ^ (2 * n) * r * r)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) * r * r = r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  do 2 rewrite <- Rmult_assoc.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² = / r ^ (2 * n) * r ^ (2 * n) * r * r
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) * r * r = r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite <- Rinv_l_sym.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r² = 1 * r * r
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) * r * r = r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold Rsqr; ring.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) * r * r = r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply pow_nonzero; assumption.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) * r * r = r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  replace (2 * S n)%nat with (S (S (2 * n))).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) * r * r = r ^ S (S (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
S (S (2 * n)) = (2 * S n)%nat
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  simpl; ring.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
S (S (2 * n)) = (2 * S n)%nat
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  ring.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rle_ge; apply pow_le; left; apply (cond_pos r).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rle_ge; apply pow_le; left; apply (cond_pos r).
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rabs_no_R0; apply pow_nonzero; assumption.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply INR_fact_neq_0.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rinv_neq_0_compat; apply INR_fact_neq_0.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (/ INR (fact (2 * n + 1))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rabs_no_R0; apply Rinv_neq_0_compat; apply INR_fact_neq_0.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rabs_no_R0; apply pow_nonzero; assumption.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply pow_nonzero; discrR.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply INR_fact_neq_0.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply pow_nonzero; discrR.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
        ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / r²
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0 + 1)) /
      ((-1) ^ n0 / INR (fact (2 * n0 + 1))))) 0 <
eps / r²
n:nat
H5:(n >= N0)%nat
H6:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n + 1)) /
      ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
eps / r²
/ INR (fact (2 * n + 1)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rinv_neq_0_compat; apply INR_fact_neq_0.
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
H0:(r : R) <> 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n + 1)) /
        ((-1) ^ n / INR (fact (2 * n + 1))))) 0 <
  eps0
eps:R
H2:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ assumption | apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption ].
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  assert (H0 := cond_pos r); red; intro; rewrite H1 in H0;
    elim (Rlt_irrefl _ H0).
Qed.